In a town whose streets form a square grid, you go from intersection A to intersection B, which is 4 blocks east and 3 blocks north of A, by a shortest route that moves only east or north. Which of the following is the number of routes?

Answer: A. 35

Every shortest route is a sequence of 7 moves: 4 one-block moves east and 3 one-block moves north. Choosing which 3 of the 7 moves go north fixes the route, so there are 7C3 = (7 × 6 × 5) ÷ (3 × 2 × 1) = 35 routes. Calculating it as permutations with identical items, 7! ÷ (4! × 3!) = 35, gives the same result.

• A. Correct. The positions of the 3 northward moves are chosen from the 7 moves, so 7C3 = 35.
• B. This is 2⁷ = 128, choosing east or north freely at each of the 7 moves, which ignores the condition of exactly 4 moves east and 3 moves north.
• C. This is 7! = 5,040, which treats all 7 moves as distinct and ignores that swapping two east moves or two north moves gives the same route.